L(s) = 1 | + 0.416i·2-s − i·3-s + 1.82·4-s − 0.478i·5-s + 0.416·6-s + 1.59i·8-s − 9-s + 0.199·10-s + (3.17 − 0.949i)11-s − 1.82i·12-s − 5.34·13-s − 0.478·15-s + 2.98·16-s + 0.246·17-s − 0.416i·18-s + 4.04·19-s + ⋯ |
L(s) = 1 | + 0.294i·2-s − 0.577i·3-s + 0.913·4-s − 0.213i·5-s + 0.170·6-s + 0.563i·8-s − 0.333·9-s + 0.0629·10-s + (0.958 − 0.286i)11-s − 0.527i·12-s − 1.48·13-s − 0.123·15-s + 0.747·16-s + 0.0597·17-s − 0.0981i·18-s + 0.927·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.843 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.225894212\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.225894212\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-3.17 + 0.949i)T \) |
good | 2 | \( 1 - 0.416iT - 2T^{2} \) |
| 5 | \( 1 + 0.478iT - 5T^{2} \) |
| 13 | \( 1 + 5.34T + 13T^{2} \) |
| 17 | \( 1 - 0.246T + 17T^{2} \) |
| 19 | \( 1 - 4.04T + 19T^{2} \) |
| 23 | \( 1 - 1.08T + 23T^{2} \) |
| 29 | \( 1 + 9.25iT - 29T^{2} \) |
| 31 | \( 1 + 1.16iT - 31T^{2} \) |
| 37 | \( 1 - 4.46T + 37T^{2} \) |
| 41 | \( 1 - 9.10T + 41T^{2} \) |
| 43 | \( 1 - 2.69iT - 43T^{2} \) |
| 47 | \( 1 + 7.86iT - 47T^{2} \) |
| 53 | \( 1 + 0.485T + 53T^{2} \) |
| 59 | \( 1 + 1.10iT - 59T^{2} \) |
| 61 | \( 1 + 5.33T + 61T^{2} \) |
| 67 | \( 1 - 3.05T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 7.88T + 73T^{2} \) |
| 79 | \( 1 - 14.2iT - 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 13.8iT - 89T^{2} \) |
| 97 | \( 1 - 8.20iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.339980085125395178164654393350, −8.302677363251946732237548206191, −7.54499577300352663929459768329, −7.02123790944539260961882723480, −6.19632733447083382065876538158, −5.46615939238984122545810286791, −4.38992754158935446639794577825, −3.03349378463392549044961617804, −2.22420057620150804349131366946, −0.962437362489928055411220725223,
1.28517302316865670087602275569, 2.60923506185419234570762047875, 3.27926439058096936746815729463, 4.41861119035709517278269711660, 5.28811529734178094007611280114, 6.31928671339612930951196634934, 7.14333773510850707845530354988, 7.61914595159916467833780650564, 8.987758696992053702467006719457, 9.534924840510523868345334134516